Exploring the Spin-Statistics Theorem: Is it a Mathematical Proof or Postulate?

[jostpuur]

Is it really a mathematical theorem or more like a "spin-statistics postulate"?

I checked the apparent proof in http://en.wikipedia.org/wiki/Spin-statistics_theorem but didn't get very convinced. If two electrons have some arbitrary spatial wave functions, you cannot switch them by rotation in general.

To me it seems, that if one quantisizes a two component complex Klein-Gordon field $\phi\in\mathbb{C}^2$, with appropriately postulated transformations with sigma matrices, one gets a theory of spin-1/2 particles that obey bose-statistics.

 

[dextercioby]

I really wonder how you could turn a dublet of complex KG fields into a spin 1/2 field... Hmm...

For a clear, technical and complete exposition of the Pauli- Lueders spin-statistics theorem see any book on axiomatical QFT. Lopuszanski, Bogolyubov, Streater & Wightman, Jost, etc.

 

[jostpuur]

I really wonder how you could turn a dublet of complex KG fields into a spin 1/2 field... Hmm...

They are not intended to be two independent scalar fields, but transformations

$$e^{-i\theta\cdot\sigma/2}$$ for rotations

and

$$e^{\eta\cdot\sigma/2}$$ for boosts

are postulated. This brings the internal angular momentum to the field. Equation of motion is just the Klein-Gordon equation for the both components separately.

 

[lsuorant]

Here's an elementary proof: http://arxiv.org/abs/1008.5382

Main points (omitting some subtleties like the massless particles and internal groups):
1) As shown by Wigner, massive spin-s one-particle states carry 2s indices of the SU(2) ("little group") fundamental representation. This is discussed also in some textbooks, although it is not standard material.
2) It is a known mathematical fact that such indices anticommute if they are permuted. I show this in the paper, and it is also mentioned in e.g. the QFT book by M. Srednicki, page 428.
3) Exchanging two one-particle states with such indices involves (2s)^2 permutations, reproducing exactly the phases given by the theorem.

Cheers,
Lauri

P.S. If there is someone who likes the paper and has publications in hep-theory, quant or math-ph, I could use endorsements to submit my other preprints.

 

[Avodyne]

I like the proof in Weinberg. Srednicki uses Weinberg's method to show that interacting, relativistic spin-0 particles must be bosons (ch.4).

lsuorant, I don't believe your proof is valid. For spin-0, it is tantamount to assuming that the wavefunction must be even on particle exchange; but this is just what you should be trying to prove! The Berry-Robbins "proof" has the same problem.

Srednicki, in ch.1, constructs a system of interacting, nonrelativistic, spin-0 fermions (see eqs.1.32 and 1.38). Since there is nothing mathematically wrong with this system, its existence demonstrates that relativity must be a necessary ingredient for a proof of the spin-statistics theorem.

 

[lsuorant]

lsuorant, I don't believe your proof is valid. For spin-0, it is tantamount to assuming that the wavefunction must be even on particle exchange; but this is just what you should be trying to prove! The Berry-Robbins "proof" has the same problem.

Spin-0 theorem is trivial. It's equivalent saying that the complex numbers commute in products. For quite some time, I thought that locality would be required to argue spin-0 commutation, as non-commutation would immediately violate it. Of course, it's a valid argument, but can be relaxed.

I don't see how Srednicki's example is connected to this. Some people seem to think that if we take a system, quantize it with the wrong relation and do not immediately arrive in a contradiction would imply that the wrong relation is "valid" in that context. That's just funny. I could invent many non-physical relations that do not immediately violate any major principle.

Spin-statistics theorem is just something that comes in the bargain of mathematical formulation of the theory.